Question: Determine how many solutions exist for the system of equations. ${8x+2y = -18}$ ${y = -4x-9}$
Solution: Convert both equations to slope-intercept form: ${8x+2y = -18}$ $8x{-8x} + 2y = -18{-8x}$ $2y = -18-8x$ $y = -9-4x$ ${y = -4x-9}$ ${y = -4x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x-9}$ ${y = -4x-9}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${8x+2y = -18}$ is also a solution of ${y = -4x-9}$, there are infinitely many solutions.